The conservation of momentum can be useful in designing control laws for underactuated mechanical systems. However, the momentum is conserved only for unactuated variables with symmetry. If a symmetry-breaking force is applied to a system, the momentum is not conserved any longer in general. The main objective of this paper is to show that there exist forces linear in velocity such as the damping force that break the symmetry but induce a new conserved quantity in place of the original momentum map. This paper formalizes that such a new conserved quantity can be constructed by combining the time integral of a general damping force and the original momentum map associated with the symmetry. Especially, we show that the new conserved quantity can exist for multiple variables with symmetry. From the perspective of stability theories, the major implication of the new conserved quantity is that the corresponding variables possess the self recovery phenomenon, i.e. they will be globally attractive to the initial condition of the variables. What is fundamental in the damping-induced self recovery is not the positivity of the damping coefficient but certain properties of the time integral of the damping force. Self recovery effect and theoretical findings are demonstrated by simulation results using a two-link manipulator and a planar pendulum. The results in this paper will be useful in designing and controlling mechanical systems with underactuation.
- Dynamic Systems and Control Division
Self Recovery Phenomenon of Mechanical Systems With Unactuated Cyclic Variables due to Damping-Like Forces
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Chang, DE, & Jeon, S. "Self Recovery Phenomenon of Mechanical Systems With Unactuated Cyclic Variables due to Damping-Like Forces." Proceedings of the ASME 2012 5th Annual Dynamic Systems and Control Conference joint with the JSME 2012 11th Motion and Vibration Conference. Volume 2: Legged Locomotion; Mechatronic Systems; Mechatronics; Mechatronics for Aquatic Environments; MEMS Control; Model Predictive Control; Modeling and Model-Based Control of Advanced IC Engines; Modeling and Simulation; Multi-Agent and Cooperative Systems; Musculoskeletal Dynamic Systems; Nano Systems; Nonlinear Systems; Nonlinear Systems and Control; Optimal Control; Pattern Recognition and Intelligent Systems; Power and Renewable Energy Systems; Powertrain Systems. Fort Lauderdale, Florida, USA. October 17–19, 2012. pp. 617-624. ASME. https://doi.org/10.1115/DSCC2012-MOVIC2012-8866
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